The most common and most likely answer to this crossword clue is the 7 letter word DRAYMAN
I've seen this crossword clue in The Times
D
R
A
Y
M
A
N
More clues leading to the result DRAYMAN
Buddy explains!

The straight part
We know that "Driver of cart" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case driver of cart = drayman.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does DRAYMAN mean?
DRAYMAN noun- A man who drives drays.
- A deliveryman for a brewery.